First, bitwise likelihood is explained. The bitwise likelihood means an index value representing, when attention is paid to a certain bit of transmission bits, the likelihood as to whether the certain bit is “0” or “1” on the reception side. In general, the bitwise likelihood is represented by a log-likelihood ratio (Formula (1)) shown below.
                              LLR          ⁡                      (                          x                              i                ,                j                                      )                          =                  log          ⁢                      {                                          P                ⁡                                  (                                                            x                                              i                        ,                        j                                                              =                                          0                      ❘                                              y                        k                                                                              )                                                            P                ⁡                                  (                                                            x                                              i                        ,                        j                                                              =                                          1                      ❘                                              y                        k                                                                              )                                                      }                                              (        1        )            
In the formula, xi,j represents a j-th bit of a symbol xi transmitted by a transmission apparatus at time i and yk represents data received by a reception apparatus at time k, which is reception data corresponding to the symbol xi transmitted at the time i. P(xi,j=0|yk) represents a conditional probability that xi,j is “0” when yk is received. P(xi,j=1|yk) represents a conditional probability that xi,j is “1” when yk is received. From the above, it can be assumed that, if LLR(xi,j) is larger than 0, xi,j is more likely to be “0” than “1” and, if LLR(xi,j) is smaller than 0, Xi,j is more likely to be “1” than “0”. Further, the magnitude of LLR(xi,j) represents the degree of the likelihood. The above Formula (1) can be represented by Formula (2) shown below when it is assumed that all transmitted bits occur with the same probability.
                              LLR          ⁡                      (                          x                              i                ,                j                                      )                          =                  log          ⁢                      {                                                            ∑                                                                                    x                        i                                            :                                              x                                                  i                          ,                          j                                                                                      =                    0                                                                                                          ⁢                                                                  ⁢                                  P                  ⁡                                      (                                                                  y                        k                                            ❘                                              x                        i                                                              )                                                                                                ∑                                                                                    x                        i                                            :                                              x                                                  i                          ,                          j                                                                                      =                    1                                                                                                          ⁢                                                                  ⁢                                  P                  ⁡                                      (                                                                  y                        k                                            ❘                                              x                        i                                                              )                                                                        }                                              (        2        )            
The numerator (a condition is xi:xi,j=0) in a log on the right side of Formula (2) is the sum of the probabilities of receiving yk when the symbol xi, the j-th bit of which is “0”, is transmitted. The denominator (a condition is xi:xi,j=1) in a log on the right side of Formula (2) is the sum of the probabilities of receiving yk when the symbol xi, the j-th bit of which is “1”, is transmitted. It is assumed that the symbol xl is transmitted, subjected to fading, and received with Gaussian noise added thereto. If correction for the fading is perfectly performed and the reception data yk is extracted on the reception side, P(yk|xi) can be approximated by Formula (3) shown below.
                              P          ⁡                      (                                          y                k                            ❘                              x                i                                      )                          ≅                              1                          2              ⁢                                                          ⁢              π              ⁢                                                          ⁢                              σ                2                                              ⁢          exp          ⁢                      {                                          -                                                      (                                                                  y                        k                                            -                                              x                        i                                                              )                                    2                                                            2                ⁢                                                                  ⁢                                  σ                  2                                                      }                                              (        3        )            
In the formula, σ2 represents the variance of the Gaussian noise and (yk−xi)2 represents the Euclidian distance between the reception data yk and the transmission symbol xi. That is, when the above Formula (3) is substituted into the formula of LLR(xi,j), the above Formula (3) can be represented by Formula (4) for the Euclidian distance between the reception data yk and the transmission symbol xi.
                              LLR          ⁡                      (                          x                              i                ,                j                                      )                          =                              log            ⁢                                          ∑                                                                            x                      i                                        :                                          x                                              i                        ,                        j                                                                              =                  0                                            ⁢                                                          ⁢                              exp                ⁢                                  {                                                            -                                                                        (                                                                                    y                              k                                                        -                                                          x                              i                                                                                )                                                2                                                                                    2                      ⁢                                                                                          ⁢                                              σ                        2                                                                              }                                                              -                      log            ⁢                                          ∑                                                                            x                      i                                        :                                          x                                              i                        ,                        j                                                                              =                  1                                            ⁢                                                          ⁢                              exp                ⁢                                  {                                                            -                                                                        (                                                                                    y                              k                                                        -                                                          x                              i                                                                                )                                                2                                                                                    2                      ⁢                                                                                          ⁢                                              σ                        2                                                                              }                                                                                        (        4        )            
In the above Formula (4), when the formula of LLR(xi,j) is approximated using only the transmission symbol xi having a minimum distance among the Euclidian distances between the reception data yk and a plurality of the transmission symbols xi, Formula (5) shown below is obtained.
                              LLR          ⁢                      (                          x                              i                ,                j                                      )                          =                                            min                                                                    x                    i                                    :                                      x                                          i                      ,                      j                                                                      =                1                                      ⁢                          {                                                (                                                            y                      k                                        -                                          x                      i                                                        )                                2                            }                                -                                    min                                                                    x                    i                                    :                                      x                                          i                      ,                      j                                                                      =                0                                      ⁢                          {                                                (                                                            y                      k                                        -                                          x                      i                                                        )                                2                            }                                                          (        5        )            
The above bitwise likelihood calculation formula is a commonly used method. As a specific likelihood calculation method, because the transmission symbol xi having a minimum Euclidian distance changes according to the reception data yk, the transmission symbols xi are classified with respect to the values of the reception data yk and different likelihood calculation formulas are retained for the respective transmission symbols xi in advance. That is, a likelihood calculation is performed by selectively using from a plurality of the retained likelihood calculation formulas according to the values of the reception data yk.
For example, Non Patent Literature 1 described below discloses a technology for reducing the computational complexity by further approximating a likelihood calculation with respect to the bitwise likelihood calculation explained above.
In the bitwise likelihood calculation explained above, it is necessary to individually retain the likelihood calculation formulas according to the values of the reception data yk. When a likelihood calculation is actually implemented in a circuit, a memory for retaining the likelihood calculation formulas is necessary. Non Patent Literature 1 described below discloses a method of approximating and simplifying likelihood calculation formulas to become one likelihood calculation formula irrespective of the values of the reception data yk. Consequently, it is possible to reduce the computational complexity and the amount of memory. Non Patent Literature 1 described below mentions that, when 16QAM (Quadrature Amplitude Modulation) and 64QAM are used as a modulation scheme and a convolutional code is used as an error correction code, then if the bit error rate obtained when the approximation of Non Patent Literature 1 is not performed and the bit error rate obtained when the approximation is performed are compared, deterioration in the bit error rate does not occur even if the approximation is performed.